# Modeling Red Blood Cell Viscosity Contrast Using Inner Soft Particle Suspension

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## Abstract

**:**

## 1. Introduction

## 2. Viscosity Contrast in Other Red Blood Cell Models

## 3. RBC Model and the Concept of Dissipative Particles

## 4. Practical Considerations for Model Implementation

## 5. Viscosity of Particle Suspensions

#### 5.1. Suspensions of Spheres or Colloids

#### 5.2. Computational Viscosity Measurements

#### 5.2.1. Shear Flow between Two Plates

#### 5.2.2. Drag Force Experiment

#### 5.3. Determining Appropriate Particle-Volume Fraction

- Random initialisation of particles inside a 2 × 2 × 2 $\mathsf{\mu}$m${}^{3}$ domain without boundaries;
- Random initialisation of particles inside a 2 × 2 × 2 $\mathsf{\mu}$m${}^{3}$ domain with boundaries in y and z direction;
- Random initialisation of particles inside red blood cell.

#### 5.4. Considerations for Initial Values of Simulation Parameters

#### 5.5. Results

## 6. Model Validation

#### 6.1. Optical Tweezers-Stretching

#### 6.2. Optical Tweezers-Release

## 7. Discussion

#### 7.1. Model Limitations

#### 7.2. Novelty

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RBC | Red blood cell |

CFL | Cell free layer |

LBM | Lattice-Boltzmann method |

IBM | Immersed boundary method |

IB-DC | Immersed boundary via dissipative coupling |

DP | Dissipative particles |

DPD | Dissipative particle dynamics |

## Appendix A

LBM Grid [$\mathsf{\mu}$m] | Time Step [$\mathsf{\mu}$s] | Viscosity [mPas] | Density [kg/m${}^{3}$] |
---|---|---|---|

1 | 0.05 | 1.2 | 1 |

**Table A2.**Simulation parameters, elastic parameters of RBC in Section 6.1 and Section 6.2 and sphere in Section 5.2.2.

RBC | Sphere | |
---|---|---|

${k}_{s}$ [$\mathsf{\mu}$N/m] | 0.009 | 2 |

${k}_{b}$ [Nm] | 0.007 | 2 |

${k}_{al}$ [$\mathsf{\mu}$N/m] | 0.09 | 2 |

${k}_{ag}$ [$\mathsf{\mu}$N/m] | 0.9 | 2 |

${k}_{v}$ [N/m${}^{2}$] | 0.9 | 2 |

no. of nodes | 1002 | 304 |

mean edge length [$\mathsf{\mu}$m] | 0.4 | 0.66 |

Soft-Sphere | Hat | |
---|---|---|

a | n | ${F}_{max}$ |

0.00032 | 1.5 | 0.6 |

**Table A4.**Simulation parameters, interaction of the particle suspension with the membrane. Outer interaction of suspension in the drag force experiment. Inner interaction of particle suspension with cell membrane from the inside, both drag force Section 5.2.2 and RBC experiments Section 6.1 and Section 6.2.

Interaction | a* | n* | ${\mathit{c}}_{\mathbf{soft}}^{*}$ [$\mathsf{\mu}$m] |
---|---|---|---|

inner | 0.02 | 1.5 | $2{r}_{h}$ |

outer | 0.128 | 1.5 | $1.25{r}_{h}$ |

**Table A5.**Effective viscosity of particle suspensions in shear flow between parallel plates. The viscosity of the underlying fluid was 1.2 mPas. For ${\gamma}_{DPD}=0$ we used the soft-sphere interaction.

$\mathit{\varphi}$ | ${\mathit{r}}_{\mathit{h}}\phantom{\rule{4pt}{0ex}}[\mathsf{\mu}$m] | ${\mathit{\gamma}}_{\mathbf{DPD}}$ [nNm${}^{-1}$s] | Shear Rate [s${}^{-1}]$ | ${\mathit{\eta}}_{\mathbf{in}}$ [mPas] |
---|---|---|---|---|

0.5 | 0.5 | 0 | 53 | 1.4 |

0.5 | 0.3 | 0 | 70 | 1.4 |

0.5 | 0.2 | 0 | 83 | 1.5 |

0.5 | 0.5 | 1.5 | 86 | 2.8 |

0.5 | 0.3 | 1.5 | 71 | 3.5 |

0.5 | 0.2 | 1.5 | 57 | 4.4 |

0.5 | 0.5 | 3 | 68 | 4.3 |

0.5 | 0.3 | 3 | 64 | 5.8 |

0.5 | 0.2 | 3 | 65 | 7.8 |

0.485 | 0.5 | 0 | 53 | 1.4 |

0.485 | 0.3 | 0 | 70 | 1.4 |

0.485 | 0.2 | 0 | 85 | 1.5 |

0.485 | 0.5 | 1.5 | 77 | 2.5 |

0.485 | 0.3 | 1.5 | 84 | 3.2 |

0.485 | 0.2 | 1.5 | 98 | 4.1 |

0.485 | 0.5 | 3 | 53 | 3.6 |

0.485 | 0.3 | 3 | 56 | 4.8 |

0.485 | 0.2 | 3 | 56 | 6.3 |

0.45 | 0.5 | 3 | 65 | 3.0 |

0.45 | 0.3 | 3 | 70 | 3.9 |

0.45 | 0.2 | 3 | 64 | 4.7 |

**Table A6.**Final axial ${d}_{a}$ and transversal ${d}_{t}$ diameters in optical tweezers stretching experiment at volume fraction $\varphi =0.5$.

${\mathit{F}}_{\mathbf{stretch}}$ [nN] | ${\mathit{d}}_{\mathit{a}}$ [$\mathsf{\mu}$m] | ${\mathit{d}}_{\mathit{t}}$ [$\mathsf{\mu}$m] |
---|---|---|

0 | 7.82 | 7.82 |

0.016 | 8.65 | 7.35 |

0.031 | 9.38 | 6.90 |

0.047 | 10.27 | 6.38 |

0.068 | 11.07 | 5.97 |

0.088 | 11.70 | 5.59 |

0.109 | 12.40 | 5.26 |

0.13 | 12.95 | 5.05 |

0.15 | 13.24 | 4.87 |

0.172 | 13.64 | 4.71 |

0.192 | 13.98 | 4.60 |

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**Figure 1.**Visualization of RBC model. (

**a**) triangular mesh modelling the membrane, (

**b**) dissipative particles included inside the membrane.

**Figure 2.**Modified computational shear flow experiment. Both upper and lower boundaries have zero no-slip conditions, with force applied to a layer of particles in the middle. This results in two mirrored sections with linear velocity profiles. (

**a**) no particles, (

**b**) particle volume fraction $\varphi =0.5$.

**Figure 3.**Initial position of a sphere in a drag force computational experiment (

**left**). A horizontal force ${F}_{drag}$ is applied to the sphere, eventually resulting in a steady terminal velocity (

**right**). To visualise the cell movement in the suspension, only a slice (width ±1 $\mathsf{\mu}$m from the central $x-y$ plane of the channel) of the suspensions is rendered. Size of the channel 96 × 96 × 96 $\mathsf{\mu}$m${}^{3}$, cell radius R = 3 $\mathsf{\mu}$m.

**Figure 4.**Boundary effect—relative deviation of computed terminal velocity from the theoretically predicted value. Sphere radius R given in $\mathsf{\mu}$m.

**Figure 5.**Viscosity for various particle radii and ${\gamma}_{DPD}$ (color coded) at $\varphi =0.5$ (solid lines) and $\varphi =0.485$ (dotted lines) in parallel plates computational experiment. For ${\gamma}_{DPD}=0$ (two gray lines that overlap) we used the soft-sphere interaction between the spherical particles. Data available in Table A5. The circles represent viscosity values from the drag force computational experiment with ${r}_{h}$ = 0.5 $\mathsf{\mu}$m. Solid circles correspond to $\varphi =0.5$ and empty circles to $\varphi =0.485$. Two grey circles almost overlap.

**Figure 6.**Relaxation of stretched RBC (${F}_{stretch}$ = 0.192 nN) to discoid shape. Dotted lines correspond to simulations with particles, solid lines represent exponential fit, either to experimental data (orange) or to simulation data (gray, black and red). $\varphi =0$ stands for simulation without inner particles.

**Table 1.**Viscosity measurements. Blood is a shear thinning fluid and thus the whole blood viscosity depends on shear rate. The values listed here approximately correspond to the shear rates 70–100 s${}^{-1}$.

Temperature [${}^{\circ}$C] | Viscosity [mPa.s = cP] | |
---|---|---|

20 | 1.00 | |

water | 25 | 0.89 |

37 | 0.69 | |

20 | 2.2 [41] | |

blood plasma | 25 | 1.63 [42] |

37 | 1.5 [41] | |

20 | ∼10 [43] | |

whole blood | 25 | ∼7 [43] |

37 | ∼5 [44] | |

RBC cytosol | 37 | 3–10 [15,45] |

PBS | 37 | 0.7 [46] |

box size L [μm] | 50 | 100 | 200 | 400 |

$R/L\phantom{\rule{4pt}{0ex}}[-]$ | 0.06 | 0.03 | 0.015 | 0.0075 |

${v}_{term}$ [mms${}^{-1}]$ | 4.59 | 5.24 | 5.63 | 5.79 |

boundary effect | $-22.02$% | $-10.95$% | $-4.44$% | $-1.61$% |

**Table 3.**Determining the appropriate particle-volume fraction $\varphi $. ${d}_{min}$ ($\mathsf{\mu}$m) is the minimal distance of two particles (over 1000 replications), ${n}_{pairs}$ is the maximum (over 1000 replications) number of particle pairs that are closer than $2{r}_{h}$, where ${r}_{h}$ = 0.2 $\mathsf{\mu}$m.

Cube, No Boundaries | Cube, Boundaries | RBC | ||||||
---|---|---|---|---|---|---|---|---|

$\varphi $ | ${d}_{min}$ | ${n}_{pairs}$ | $\varphi $ | ${d}_{min}$ | ${n}_{pairs}$ | $\varphi $ | ${d}_{min}$ | ${n}_{pairs}$ |

0.545 | 0.400 | 0 | 0.545 | 0.351 | 93 | 0.503 | 0.371 | 513 |

0.503 | 0.400 | 0 | 0.524 | 0.39994 | 3 | 0.485 | 0.389 | 40 |

0.461 | 0.400 | 0 | 0.503 | 0.400 | 0 | 0.467 | 0.400 | 0 |

**Table 4.**Comparison of effective viscosity [mPas] of particle suspensions in shear flow between parallel plates vs. in drag force experiment. The viscosity of the underlying fluid was 1.2 mPas, ${\gamma}_{DPD}$ = 3 nNm${}^{-1}$s and the radius of DPs was ${r}_{h}$ = 0.5 $\mathsf{\mu}$m, shear rate given in $\left[{s}^{-1}\right]$.

Parallel Plates | Drag Force Sphere | |||
---|---|---|---|---|

$\varphi $ | shear rate | ${\eta}_{in}$ | shear rate | ${\eta}_{in}$ |

0.5 | 68 | 4.3 | 39 | 4.1 |

0.485 | 53 | 3.6 | 41 | 3.7 |

0.45 | 65 | 3.0 | 58 | 2.9 |

**Table 5.**At low shear rates, the shear rate has almost no impact on the particle suspension viscosity (here $\varphi =0.5$).

${\mathit{r}}_{\mathit{h}}$ | ${\mathit{\gamma}}_{\mathbf{DPD}}$ | Shear Rate | ${\mathit{\eta}}_{\mathbf{in}}$ |
---|---|---|---|

[$\mathsf{\mu}$m] | [nNm${}^{-1}$s] | [s${}^{-1}$] | [mPas] |

0.5 | 3.0 | 27 | 4.4 |

0.5 | 3.0 | 68 | 4.3 |

0.3 | 3.0 | 36 | 5.7 |

0.3 | 3.0 | 64 | 5.8 |

0.2 | 1.5 | 57 | 4.4 |

0.2 | 1.5 | 113 | 4.4 |

**Table 6.**Decrease in effective viscosity of particle suspension ${\eta}_{in}$ with respect to the decreasing volume fraction $\varphi $. Here ${\gamma}_{DPD}$ = 3 nNm${}^{-1}$s, radii given in $\mathsf{\mu}$m.

Viscosity [mPas] | |||
---|---|---|---|

$\mathbf{\varphi}$ | ${\mathbf{r}}_{\mathbf{h}}=\mathbf{0}.\mathbf{2}$ | ${\mathbf{r}}_{\mathbf{h}}=\mathbf{0}.\mathbf{3}$ | ${\mathbf{r}}_{\mathbf{h}}=\mathbf{0}.\mathbf{5}$ |

0.5 | 7.8 | 5.8 | 4.3 |

0.485 | 6.3 | 4.8 | 3.6 |

0.45 | 4.7 | 3.9 | 3.0 |

**Table 7.**Parameters of the inner particle suspension for RBC that result in appropriate effective viscosity ${\eta}_{in}$.

$\mathit{\varphi}$ | ${\mathit{r}}_{\mathit{h}}$ | ${\mathit{\gamma}}_{\mathbf{DPD}}$ | ${\mathit{\eta}}_{\mathbf{in}}$ |
---|---|---|---|

$[-]$ | [$\mathsf{\mu}$m] | [nNm${}^{-1}$s] | [mPas] |

0.5 | 0.2 | 2.5 | 6.0 |

0.5 | 0.3 | 3.4 | 6.0 |

0.485 | 0.2 | 2.75 | 6.0 |

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**MDPI and ACS Style**

Bohiniková, A.; Jančigová, I.; Cimrák, I.
Modeling Red Blood Cell Viscosity Contrast Using Inner Soft Particle Suspension. *Micromachines* **2021**, *12*, 974.
https://doi.org/10.3390/mi12080974

**AMA Style**

Bohiniková A, Jančigová I, Cimrák I.
Modeling Red Blood Cell Viscosity Contrast Using Inner Soft Particle Suspension. *Micromachines*. 2021; 12(8):974.
https://doi.org/10.3390/mi12080974

**Chicago/Turabian Style**

Bohiniková, Alžbeta, Iveta Jančigová, and Ivan Cimrák.
2021. "Modeling Red Blood Cell Viscosity Contrast Using Inner Soft Particle Suspension" *Micromachines* 12, no. 8: 974.
https://doi.org/10.3390/mi12080974